Objective This proposal aims to unravel mysteries at the frontier of number theory and other areas of mathematics and physics. The main focus will be to understand and exploit “modularity” of q-hypergeometric series. “Modular forms are functions on the complex plane that are inordinately symmetric.” (Mazur) The motivation comes from the wide-reaching applications of modularity in combinatorics, percolation, Lie theory, and physics (black holes).The interplay between automorphic forms, q-series, and other areas of mathematics and physics is often two-sided. On the one hand, the other areas provide interesting examples of automorphic objects and predict their behavior. Sometimes these even motivate new classes of automorphic objects which have not been previously studied. On the other hand, knowing that certain generating functions are modular gives one access to deep theoretical tools to prove results in other areas. “Mathematics is a language, and we need that language to understand the physics of our universe.”(Ooguri) Understanding this interplay has attracted attention of researchers from a variety of areas. However, proofs of modularity of q-hypergeometric series currently fall far short of a comprehensive theory to describe the interplay between them and automorphic forms. A recent conjecture of W. Nahm relates the modularity of such series to K-theory. In this proposal I aim to fill this gap and provide a better understanding of this interplay by building a general structural framework enveloping these q-series. For this I will employ new kinds of automorphic objects and embed the functions of interest into bigger familiesA successful outcome of the proposed research will open further horizons and also answer open questions, even those in other areas which were not addressed in this proposal; for example the new theory could be applied to better understand Donaldson invariants. Fields of science natural sciencesmathematicspure mathematicsarithmeticsnatural sciencesphysical sciencesastronomyastrophysicsblack holesnatural sciencesmathematicspure mathematicsdiscrete mathematicscombinatorics Programme(s) FP7-IDEAS-ERC - Specific programme: "Ideas" implementing the Seventh Framework Programme of the European Community for research, technological development and demonstration activities (2007 to 2013) Topic(s) ERC-SG-PE1 - ERC Starting Grant - Mathematical foundations Call for proposal ERC-2013-StG See other projects for this call Funding Scheme ERC-SG - ERC Starting Grant Host institution UNIVERSITAT ZU KOLN EU contribution € 1 240 500,00 Address ALBERTUS MAGNUS PLATZ 50931 Koln Germany See on map Region Nordrhein-Westfalen Köln Köln, Kreisfreie Stadt Activity type Higher or Secondary Education Establishments Administrative Contact Caren Wiegand (Mrs.) Principal investigator Kathrin Bringmann (Prof.) Links Contact the organisation Opens in new window Website Opens in new window Total cost No data Beneficiaries (1) Sort alphabetically Sort by EU Contribution Expand all Collapse all UNIVERSITAT ZU KOLN Germany EU contribution € 1 240 500,00 Address ALBERTUS MAGNUS PLATZ 50931 Koln See on map Region Nordrhein-Westfalen Köln Köln, Kreisfreie Stadt Activity type Higher or Secondary Education Establishments Administrative Contact Caren Wiegand (Mrs.) Principal investigator Kathrin Bringmann (Prof.) Links Contact the organisation Opens in new window Website Opens in new window Total cost No data
